arXiv 2402.010062024-02-01math.CO

图的边映射：Turán型参数

Edge mappings of graphs: Turán type parameters

Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer

AI总结：

本文研究图的边映射相关Turán型参数，确定了匹配图在大顶点数下的最大边数参数值，刻画了森林的极小不可避免图集合的有限性条件，并提出非森林图对应集合无限的猜想。

摘要：

本文研究与图的边映射相关的参数问题。我们定义量$h(n,G)$为$n$顶点图$H$的最大边数，满足存在映射$f: E(H)\rightarrow E(H)$，对所有$e\in E(H)$有$f(e)\neq e$，且进一步地，在$H$中所有$G$的拷贝$G'$中，存在$e\in E(G')$使得$f(e)\in E(G')$。在诸多结果中，我们确定了当$G$为匹配且$n$足够大时的$h(n,G)$值。作为相关概念，若对任意满足$f(e)\neq e$的映射$f: E(H)\rightarrow E(H)$，$H$中都存在$G$的拷贝$G'$，使得对所有$e\in E(G')$有$f(e)\notin E(G')$，则称$H$对$G$是不可避免的。$G$的极小不可避免图集合记为$\mathcal{M}(G)$。我们证明：若$F$是森林，则$\mathcal{M}(F)$有限当且仅当$F$是匹配；并猜想对所有非森林图$G$，集合$\mathcal{M}(G)$是无限的。本文还定义了若干其他参数并证明了基础结果，仍有大量未解决问题。

英文摘要：

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ for all $e\in E$ and further in all copies $G'$ of $G$ in $H$ there exists $e\in E(G')$ with $f(e)\in E(G')$. Among other results, we determine $h(n, G)$ when $G$ is a matching and $n$ is large enough. As a related concept, we say that $H$ is unavoidable for $G$ if for any mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ there exists a copy $G'$ of $G$ in $H$ such that $f(e)\notin E(G')$ for all $e\in E(G)$. The set of minimal unavoidable graphs for $G$ is denoted by $\mathcal{M}(G)$. We prove that if $F$ is a forest, then $\mathcal{M}(F)$ is finite if and only if $F$ is a matching, and we conjecture that for all non-forest graphs $G$, the set $\mathcal{M}(G)$ is infinite. Several other parameters are defined with basic results proved. Lots of open problems remain.

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